Subgroups — #26032797

Order 11392.

348 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152294	C₂	1	1	yes	maximal, characteristic
2	#19152004	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151936	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151926	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151922	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151922	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151905	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151885	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151846	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151829	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151696	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151670	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151641	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151634	C₂	1	1	yes	maximal, characteristic, complemented
15	#556×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083561	C₂²	1	1	yes	characteristic
17	C₂×#13083558	C₂²	1	1	yes	characteristic
18	C₂²×#10623494	C₂²	1	1	yes	characteristic
19	#15668007	—	2	1	no	—
20	#15668004	C₂²	1	1	yes	characteristic
21	#15667994	—	2	1	no	—
22	#15667986	C₂²	1	1	yes	characteristic
23	#15667986	C₂²	1	1	yes	characteristic
24	#15667984	—	2	1	no	—
25	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
26	#15667972	—	2	1	no	—
27	#15667971	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
29	#15667945	C₂²	1	1	yes	characteristic
30	#15667945	C₂²	1	1	yes	characteristic
31	#15667945	C₂²	1	1	yes	characteristic
32	#15667945	C₂²	1	1	yes	characteristic
33	#15667943	C₂²	1	1	yes	characteristic
34	#15667943	C₂²	1	1	yes	characteristic
35	#15667943	C₂²	1	1	yes	characteristic
36	#15667943	C₂²	1	1	yes	characteristic
37	#15667943	C₂²	1	1	yes	characteristic
38	C₄⋊#10623494	C₂²	1	1	yes	characteristic
39	#15667936	C₂²	1	1	yes	characteristic
40	#15667936	C₂²	1	1	yes	characteristic
41	#15667936	C₂²	1	1	yes	characteristic
42	#15667932	C₂²	1	1	yes	characteristic
43	#15667932	C₂²	1	1	yes	characteristic
44	#15667932	C₂²	1	1	yes	characteristic
45	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
46	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
47	#15667923	C₂²	1	1	yes	characteristic
48	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
49	#126×C₈₉	—	2	1	no	—
50	#126×C₈₉	—	2	1	no	—
51	#124×C₈₉	—	2	1	no	—
52	C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
53	C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₈₉	—	2	1	no	—
55	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
56	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
57	#117×C₈₉	C₂²	1	1	yes	characteristic
58	#117×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊D₈₉	—	2	1	no	—
61	C₂×C₂²⋊D₈₉	—	2	1	no	—
62	C₂²×Dic₈₉	—	2	1	no	—
63	C₂×C₄×D₈₉	—	2	1	no	—
64	#13083561	—	2	2	no	—
65	#13083558	—	2	2	no	—
66	C₂×#10623494	—	2	2	no	—
67	C₂×#10623494	C₂³	1	1	yes	characteristic
68	C₂×#10623494	C₂³	1	1	yes	characteristic
69	C₂×#10623494	C₂³	1	1	yes	characteristic
70	C₂×#10623494	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₈₉	—	2	1	no	—
72	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
75	#13083553	—	2	1	no	—
76	#13083553	—	2	1	no	—
77	C₂²⋊Dic₈₉	—	2	1	no	—
78	#13083551	—	2	1	no	—
79	#13083551	—	2	1	no	—
80	#13083551	—	2	1	no	—
81	C₄×Dic₈₉	—	2	1	no	—
82	C₂×Q₈×C₈₉	—	2	1	no	—
83	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
84	C₂×D₄×C₈₉	D₄	1	1	yes	characteristic
85	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
86	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
87	C₄⋊C₄×C₈₉	—	2	1	no	—
88	C₄⋊C₄×C₈₉	—	2	1	no	—
89	C₄⋊C₄×C₈₉	—	2	1	no	—
90	C₄⋊C₄×C₈₉	—	2	1	no	—
91	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
92	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
93	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
94	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
95	C₂²⋊C₄×C₈₉	—	2	1	no	—
96	C₂²⋊C₄×C₈₉	—	2	1	no	—
97	C₂²⋊C₄×C₈₉	—	2	1	no	—
98	C₂²⋊C₄×C₈₉	—	2	1	no	—
99	C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
100	C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
101	C₄²×C₈₉	—	2	1	no	—
102	C₄²×C₈₉	—	2	1	no	—
103	C₄²×C₈₉	—	2	1	no	—
104	C₄²×C₈₉	C₂³	1	1	yes	characteristic
105	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₂²×D₈₉	—	2	1	no	—
108	Q₈×C₈₉	—	2	1	no	—
109	Q₈×C₈₉	—	2	1	no	—
110	Q₈×C₈₉	—	4	2	no	—
111	D₄×C₈₉	—	4	1	no	—
112	C₂×C₄×C₈₉	—	2	1	no	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
125	C₂²⋊D₈₉	—	4	2	no	—
126	C₂²⋊D₈₉	—	8	1	no	—
127	C₂×Dic₈₉	—	2	1	no	—
128	C₂×Dic₈₉	—	2	1	no	—
129	C₂×Dic₈₉	—	2	1	no	—
130	C₂×Dic₈₉	—	8	1	no	—
131	D₃₅₆	—	2	1	no	—
132	D₃₅₆	—	2	1	no	—
133	C₄×D₈₉	—	8	1	no	—
134	#10623494	—	2	1	no	—
135	#10623494	—	2	1	no	—
136	#10623494	—	2	1	no	—
137	#10623494	#41	1	2	yes	—
138	C₂²×C₈₉	—	2	1	no	—
139	C₂²×C₈₉	—	2	1	no	—
140	C₂²×C₈₉	—	2	1	no	—
141	C₂²×C₈₉	—	2	1	no	—
142	C₂²×C₈₉	—	2	1	no	—
143	C₂²×C₈₉	—	2	1	no	—
144	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₈₉	—	4	1	no	—
146	C₂×D₈₉	—	4	1	no	—
147	C₂×D₈₉	—	4	1	no	—
148	C₄×C₈₉	—	2	1	no	—
149	C₄×C₈₉	—	2	1	no	—
150	C₄×C₈₉	—	2	1	no	—
151	C₄×C₈₉	—	2	1	no	—
152	C₄×C₈₉	—	4	1	no	—
153	C₄×C₈₉	—	4	1	no	—
154	C₄×C₈₉	—	4	1	no	—
155	C₄×C₈₉	—	4	1	no	—
156	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
157	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
158	Dic₈₉	—	4	1	no	—
159	Dic₈₉	—	4	1	no	—
160	Dic₈₉	—	4	1	no	—
161	C₂×C₈₉	—	4	1	no	—
162	C₂×C₈₉	—	4	1	no	—
163	C₂×C₈₉	#547	1	1	yes	characteristic
164	C₂×C₈₉	#575	1	1	yes	characteristic
165	C₂×C₈₉	#577	1	1	yes	characteristic
166	D₈₉	—	8	1	no	—
167	#3350	—	89	1	no	maximal, Sylow
168	C₈₉	#3350	1	1	yes	characteristic, complemented, Sylow
169	#556	D₈₉	1	1	yes	characteristic, complemented
170	#553	—	89	1	no	—
171	#501	—	89	1	no	—
172	#485	—	89	1	no	—
173	#483	—	89	1	no	—
174	#482	—	89	1	no	—
175	#482	—	89	1	no	—
176	#479	—	89	1	no	—
177	#476	—	89	1	no	—
178	#468	—	89	1	no	—
179	#465	—	89	1	no	—
180	#440	—	89	1	no	—
181	#435	—	89	1	no	—
182	#428	—	89	1	no	—
183	#427	—	89	1	no	—
184	C₂×SD₁₆	—	89	1	no	—
185	C₂×M₁₆	—	89	1	no	—
186	C₂²×C₈	—	89	1	no	—
187	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
188	#126	—	2	1	no	—
189	#126	—	2	1	no	—
190	#126	—	178	1	no	—
191	#125	—	89	1	no	—
192	#124	—	2	1	no	—
193	#123	—	178	1	no	—
194	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
195	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
196	#121	—	89	1	no	—
197	#121	—	89	1	no	—
198	#121	—	178	1	no	—
199	C₄×Q₈	—	2	1	no	—
200	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
201	C₄×D₄	—	89	1	no	—
202	C₄×D₄	—	178	1	no	—
203	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
204	#117	—	89	1	no	—
205	#117	C₂×D₈₉	1	1	yes	characteristic
206	#117	C₂×D₈₉	1	1	yes	characteristic
207	C₂×C₄⋊C₄	—	89	1	no	—
208	#107	—	89	1	no	—
209	#107	—	89	1	no	—
210	#107	—	89	1	no	—
211	#107	—	89	1	no	—
212	#106	—	89	1	no	—
213	#106	—	89	1	no	—
214	#106	—	89	1	no	—
215	#106	—	89	1	no	—
216	#106	—	89	1	no	—
217	C₄⋊C₈	—	89	1	no	—
218	#103	—	89	1	no	—
219	#103	—	89	1	no	—
220	#103	—	89	1	no	—
221	#102	—	89	1	no	—
222	#102	—	89	1	no	—
223	#102	—	89	1	no	—
224	C₂²⋊C₈	—	89	1	no	—
225	C₂²⋊C₈	—	89	1	no	—
226	#97	—	89	1	no	—
227	C₂×Q₈	—	2	1	no	—
228	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
229	C₂×D₄	—	89	1	no	—
230	C₂×D₄	—	178	1	no	—
231	C₂×D₄	—	178	1	no	—
232	C₂×D₄	C₂²⋊D₈₉	1	1	yes	characteristic
233	C₂²×C₄	—	178	1	no	—
234	C₂²×C₄	—	178	1	no	—
235	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
236	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
237	SD₁₆	—	178	2	no	—
238	M₁₆	—	178	2	no	—
239	C₂×C₈	—	89	1	no	—
240	C₂×C₈	—	89	1	no	—
241	C₂×C₈	—	89	1	no	—
242	C₂×C₈	—	89	1	no	—
243	C₂×C₈	—	178	2	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	89	1	no	—
249	C₄⋊C₄	—	89	1	no	—
250	C₄⋊C₄	—	89	1	no	—
251	C₄⋊C₄	—	178	1	no	—
252	C₄⋊C₄	—	178	1	no	—
253	C₄⋊C₄	—	178	1	no	—
254	C₄⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
257	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	178	1	no	—
263	C₂²⋊C₄	—	178	1	no	—
264	C₂²⋊C₄	—	178	1	no	—
265	C₂²⋊C₄	—	178	1	no	—
266	C₂²⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
267	C₂²⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
268	C₄²	—	2	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	178	1	no	—
272	C₄²	C₂²×D₈₉	1	1	yes	characteristic
273	C₂³	—	178	1	no	—
274	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
275	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	4	2	no	—
279	D₄	—	4	1	no	—
280	D₄	—	356	2	no	—
281	D₄	—	178	1	no	—
282	D₄	—	178	1	no	—
283	D₄	—	712	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	178	1	no	—
293	C₂×C₄	—	178	1	no	—
294	C₂×C₄	—	178	1	no	—
295	C₂×C₄	—	712	1	no	—
296	C₂×C₄	—	712	1	no	—
297	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
298	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
299	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
300	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
301	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
302	C₈	—	89	2	no	—
303	C₈	—	178	1	no	—
304	C₈	—	178	1	no	—
305	C₈	—	178	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	356	1	no	—
313	C₂²	—	356	1	no	—
314	C₂²	—	356	1	no	—
315	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	356	1	no	—
325	C₄	—	356	1	no	—
326	C₄	—	356	1	no	—
327	C₄	#41×D₈₉	1	1	yes	characteristic
328	C₄	#15668059	1	1	yes	characteristic
329	C₂	—	4	1	no	—
330	C₂	—	4	1	no	—
331	C₂	—	712	1	no	—
332	C₂	#19152247	1	1	yes	characteristic
333	C₂	#19152396	1	1	yes	characteristic
334	C₂	#19152406	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#556×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152294	C₂
hypercenter	#556	D₈₉
hypocenter	C₈₉	#3350
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #556

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #19152004	C₂	semidirect
3. #19151936	C₂	semidirect
4. #19151926	C₂	semidirect
5. #19151922	C₂	semidirect
6. #19151922	C₂	semidirect
7. #19151905	C₂	semidirect
8. #19151885	C₂	semidirect
9. #19151846	C₂	semidirect
10. #19151829	C₂	semidirect
11. #19151696	C₂	semidirect
12. #19151670	C₂	semidirect
13. #19151641	C₂	semidirect
14. #19151634	C₂	semidirect
15. #556×C₈₉	C₂	semidirect
168. C₈₉	#3350	semidirect
169. #556	D₈₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#19152294	2
#19152004	2
#19151936	2
#19151926	2
#19151922	2
#19151905	2
#19151885	2
#19151846	2
#19151829	2
#19151696	2
#19151670	2
#19151641	2
#19151634	2
#556×C₈₉	2
#3350	89